LMFDB - Newform orbit 350.4.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,4,Mod(1,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$350 = 2 \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,2,4,0,-4,-7,-8,-23,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$20.6506685020$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

f(q)

=

q - 2 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{6} - 7 q^{7} - 8 q^{8} - 23 q^{9} + 48 q^{11} + 8 q^{12} - 56 q^{13} + 14 q^{14} + 16 q^{16} + 114 q^{17} + 46 q^{18} + 2 q^{19} - 14 q^{21} - 96 q^{22} + 120 q^{23}+ \cdots - 1104 q^{99}+O(q^{100})

q - 2 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^6 - 7 * q^7 - 8 * q^8 - 23 * q^9 + 48 * q^11 + 8 * q^12 - 56 * q^13 + 14 * q^14 + 16 * q^16 + 114 * q^17 + 46 * q^18 + 2 * q^19 - 14 * q^21 - 96 * q^22 + 120 * q^23 - 16 * q^24 + 112 * q^26 - 100 * q^27 - 28 * q^28 - 54 * q^29 + 236 * q^31 - 32 * q^32 + 96 * q^33 - 228 * q^34 - 92 * q^36 - 146 * q^37 - 4 * q^38 - 112 * q^39 + 126 * q^41 + 28 * q^42 + 376 * q^43 + 192 * q^44 - 240 * q^46 + 12 * q^47 + 32 * q^48 + 49 * q^49 + 228 * q^51 - 224 * q^52 - 174 * q^53 + 200 * q^54 + 56 * q^56 + 4 * q^57 + 108 * q^58 + 138 * q^59 + 380 * q^61 - 472 * q^62 + 161 * q^63 + 64 * q^64 - 192 * q^66 + 484 * q^67 + 456 * q^68 + 240 * q^69 + 576 * q^71 + 184 * q^72 + 1150 * q^73 + 292 * q^74 + 8 * q^76 - 336 * q^77 + 224 * q^78 + 776 * q^79 + 421 * q^81 - 252 * q^82 - 378 * q^83 - 56 * q^84 - 752 * q^86 - 108 * q^87 - 384 * q^88 - 390 * q^89 + 392 * q^91 + 480 * q^92 + 472 * q^93 - 24 * q^94 - 64 * q^96 + 1330 * q^97 - 98 * q^98 - 1104 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

−2.00000

2.00000

4.00000

−4.00000

−7.00000

−8.00000

−23.0000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$7$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	350.4.a.f		1
5.b	even	2	1		14.4.a.b	✓	1
5.c	odd	4	2		350.4.c.g		2
7.b	odd	2	1		2450.4.a.i		1
15.d	odd	2	1		126.4.a.d		1
20.d	odd	2	1		112.4.a.e		1
35.c	odd	2	1		98.4.a.e		1
35.i	odd	6	2		98.4.c.b		2
35.j	even	6	2		98.4.c.c		2
40.e	odd	2	1		448.4.a.g		1
40.f	even	2	1		448.4.a.k		1
55.d	odd	2	1		1694.4.a.b		1
60.h	even	2	1		1008.4.a.r		1
65.d	even	2	1		2366.4.a.c		1
105.g	even	2	1		882.4.a.b		1
105.o	odd	6	2		882.4.g.p		2
105.p	even	6	2		882.4.g.v		2
140.c	even	2	1		784.4.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.a.b	✓	1	5.b	even	2	1
98.4.a.e		1	35.c	odd	2	1
98.4.c.b		2	35.i	odd	6	2
98.4.c.c		2	35.j	even	6	2
112.4.a.e		1	20.d	odd	2	1
126.4.a.d		1	15.d	odd	2	1
350.4.a.f		1	1.a	even	1	1	trivial
350.4.c.g		2	5.c	odd	4	2
448.4.a.g		1	40.e	odd	2	1
448.4.a.k		1	40.f	even	2	1
784.4.a.h		1	140.c	even	2	1
882.4.a.b		1	105.g	even	2	1
882.4.g.p		2	105.o	odd	6	2
882.4.g.v		2	105.p	even	6	2
1008.4.a.r		1	60.h	even	2	1
1694.4.a.b		1	55.d	odd	2	1
2366.4.a.c		1	65.d	even	2	1
2450.4.a.i		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(350))$ :

T_{3} - 2

T3 - 2

T_{11} - 48

T11 - 48

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 2$ T + 2
$3$	$T - 2$ T - 2
$5$	$T$ T
$7$	$T + 7$ T + 7
$11$	$T - 48$ T - 48
$13$	$T + 56$ T + 56
$17$	$T - 114$ T - 114
$19$	$T - 2$ T - 2
$23$	$T - 120$ T - 120
$29$	$T + 54$ T + 54
$31$	$T - 236$ T - 236
$37$	$T + 146$ T + 146
$41$	$T - 126$ T - 126
$43$	$T - 376$ T - 376
$47$	$T - 12$ T - 12
$53$	$T + 174$ T + 174
$59$	$T - 138$ T - 138
$61$	$T - 380$ T - 380
$67$	$T - 484$ T - 484
$71$	$T - 576$ T - 576
$73$	$T - 1150$ T - 1150
$79$	$T - 776$ T - 776
$83$	$T + 378$ T + 378
$89$	$T + 390$ T + 390
$97$	$T - 1330$ T - 1330
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